Gamma-convergence of gradient flows on Hilbert and metric spaces and  applications

Serfaty, Sylvia

doi:10.3934/dcds.2011.31.1427

Cited by 149 publications

(224 citation statements)

References 42 publications

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“…Ambrosio and Serfaty [1] and Ambrosio, Mainini, and Serfaty [2] study them as a metric gradient flow in the space of measures with the Wasserstein distance as the natural metric; however, they do not obtain the convergence. Even when it becomes possible to carry out the program outlined in the survey of Serfaty [47] and to directly obtain the Wasserstein gradient flow studied in [1, 2] from the Gorkov-Eliashberg equation by the Γ -convergence of a gradient flow type result, we believe that our approach will still be useful. On the one hand, it provides quantitative bounds that are useful in type II superconductors without going to the ε → 0 limit of "extreme" type II superconductivity.…”

Section: Results In the Following We Letmentioning

confidence: 99%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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We establish vortex dynamics for the time-dependent Ginzburg-Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

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Section: Results In the Following We Letmentioning

confidence: 99%

Vortex Liquids and the Ginzburg–landau Equation

Kurzke

Spirn

2014

Forum math. Sigma

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“…Variational-evolution structure, such as in the case of gradient flows and variational rate-independent systems, also facilitates limits [28,51,53,54,67,70,71].…”

Section: Introductionmentioning

confidence: 99%

Variational approach to coarse-graining of generalized gradient flows

et al. 2017

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In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with nondissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the smallnoise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom. Mathematics Subject Classification

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“…Sandier et al [38,40] appear to be the first to explore in detail the use of this structure for passing to the limit. Serfaty [39] discusses the case of metric spaces, with obvious applications for the case of the Wasserstein metric. She leaves aside the question of compactness, however, and one of the main contributions of this paper is to show that appropriate compactness 'in time' can also be obtained from the Wasserstein structure.…”

Section: Discussionmentioning

confidence: 99%

“…Therefore one cannot canonically separate these two contributions in the structure of J 0 . This fact has another interesting consequence: in the present setting it is not possible to investigate separately the limit behaviour of the distance and of the functional using -convergence tools (as in the well-behaved gradient flows considered by [4,35,37,39]). Conversely, the geometry perturbed by the sublevels and by the slopes of the varying entropy functionals E free ε induces a new kind of evolution in the limit, which can solely be captured by considering the asymptotics of the whole space-time functionals J ε .…”

Section: The Variational Approach: Basic Tools and Main Ideasmentioning

confidence: 99%

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Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

et al. 2011

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We study a singular-limit problem arising in the modelling of chemical reactions. At finite ε > 0, the system is described by a Fokker-Planck convection-diffusion equation with a double-well convection potential. This potential is scaled by 1/ε, and in the limit ε → 0, the solution concentrates onto the two wells, resulting into a limiting system that is a pair of ordinary differential equations for the density at the two wells. This convergence has been proved in Peletier et al. (SIAM J Math Anal, 42(4):1805-1825, 2010, using the linear structure of the equation. In this study we re-prove the result by using solely the Wasserstein gradient-flow structure of the system. In particular we make no use of the linearity, nor of the fact that it is a second-order system. The first key step in this approach is a reformulation of the equation as the minimization of an action functional that captures the property of being a curve of maximal slope in an integrated form. The second important step is a rescaling of space. Using only the Wasserstein gradient-flow structure, we prove that the sequence of rescaled solutions is pre-compact in an appropriate topology. We then prove a Gamma-convergence result for the functional in this topology, and we identify the limiting functional and the differential equation that it represents. A consequence of these results is that solutions of the ε-problem converge to a solution of the limiting problem.

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Gamma-convergence of gradient flows on Hilbert and metric spaces and applications

Cited by 149 publications

References 42 publications

Vortex Liquids and the Ginzburg–landau Equation

Vortex Liquids and the Ginzburg–landau Equation

Variational approach to coarse-graining of generalized gradient flows

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

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